Modular arithmetic identifies integers that have the same remainder after division by a fixed modulus.
For a positive modulus , a reduced residue system contains exactly one representative of each residue class coprime to . Its size is the Euler totient function . Multiplication by any unit modulo n permutes these classes, which proves the Fermat-Euler theorem after multiplying and cancelling their unit product.
A residue class modulo is a unit exactly when it is represented by an integer coprime to . The units form the multiplicative group .
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Modular arithmetic, often referred to as "clock arithmetic," is a system of arithmetic for integers, where numbers wrap around after reaching a certain value known as the modulus. In modular arithmetic, two numbers are considered equivalent if they have the same remainder when divided by the modulus. The basic notation for modular arithmetic is \( a \equiv b \mod m \), which means that \( a \) and \( b \) give the same remainder when divided by \( m \).
